Geodesics, extension of holomorphic functions and the spectral theory of multioperators

One of the most successful and beautiful branches of mathematics in the nineteenth and twentieth centuries was the theory of analytic functions. These are functions which are smooth enough to have a gradient at every point of a region of the complex plane. The theory has had enormous importance for the understanding of several branches of physics and engineering, as well as playing an essential role in pure mathematics. Since the 1920s or thereabouts there has been an analogous development of a theory of analytic functions of several variables, which is also significant for science and technology.

In particular, some engineering design problems require the construction of functions of a single variable which are rational (that is, expressible by a formula involving only addition, multiplication and division), take their values in some prescribed target region of higher-dimensional complex space and meet some further specifications. For certain special target regions there is a well-developed theory already; this theory plays a significant role in `H infinity control', a branch of control engineering.

The present project will provide the opportunity for the PI to study new operator-theoretic methods at the University of California at San Diego, mainly, from Professor Jim Agler. His deep understanding of several complex variables and operator-theoretic methods are vital for the project. The proposed research will extend existing theory to include other target regions of engineering relevance, building on discoveries about the geometry and function theory of such regions by many mathematicians. One of our principal aims is to develop a theory of rational functions from a disc or half-plane to regions such as the symmetrised bidisc which parallels the classical theory. There are substantial difficulties in carrying out such a development: here are three of them. Firstly, whereas the classical target regions are homogeneous, meaning that any point is like any other, the symmetrised bidisc is inhomogeneous, so that some points have special geometric properties. Secondly, whereas classical domains are convex, the symmetrised bidisc is not even isomorphic to a convex domain. Furthermore, the symmetrised bidisc has sharp corners, for which reason many of the results of mainstream several complex variables do not apply to it.

Nevertheless, research over the past decade has shown that the symmetrised bidisc and some similar domains have a rich geometry and function theory, exhibiting fascinating new features that do not appear in classical domains. We shall exploit the close connection between the symmetrised bidisc and two classical domains (the bidisc and the unit ball of the space of 2 x 2 matrices) to identify sets in the symmetrized bidisc with the norm-preserving extension property and to get new properties of $\Gamma$-contractions. We intend to do likewise for other target domains, which we call quasi-Cartan domains, to indicate their close connection with the classical `Cartan domains'.

The results of the project will be significant for researchers in several complex variables and in the theory of linear operators; there are many of both categories worldwide. They will also be significant for control engineers, particularly those who use the technique of `mu-synthesis' for the design of automatic controllers for linear plants subject to structured uncertainty.

In the first instance this research will benefit to specialists in two substantial branches of mathematical analysis, namely, operator theory and several complex variables. Complex variable theory is fundamental to several important topics in physics and engineering, notably electrical circuits, quantum mechanics, systems theory and control theory. Classically it is functions of a single complex variable that are most important, but there are also more recent applications which require the theory of analytic functions of several variables. Examples are multi-dimensional circuit theory and control theory. In the latter application the extra variables typically represent uncertainties in the plant model. Earlier work of Agler and Young have shown how the analysis of the function theory of certain domains in higher-dimensional complex space can lead to the solution of special cases of the ``mu-synthesis problem", an important, hard problem in control engineering. The present project will lead to a better understanding of analytic functions on a broad class of higher-dimensional domains, and hence potentially to applications in physics and engineering. In particular it has a bearing on the control of systems with structured uncertainty. This is a significant problem in "H infinity control", a branch of control engineering that has been applied within the past two decades to a wide range of products, including automatic pilots for aircraft, wind farms and CD reading heads, among many other devices. Accordingly the outcome of the proposed research may in the longer term play a part in an important technology.